Numerical Solution of Distributed-Order Fractional Diffusion Equations Using a High-Order Temporal Scheme

Abstract

Distributed-order fractional diffusion equations (DO-FDEs) are crucial for modeling complex processes in heterogeneous and anomalous systems. Unfortunately, they encounter significant analytical and computational challenges. This study introduces a novel numerical framework that extends high-order approximation formulas to accommodate the distributed-order fractional derivative. The framework achieves a temporal convergence order of

4-\beta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$4 - \beta $$\end{document}

, where

\beta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}

is the upper bound of the integral in the distributed-order derivative. The proposed scheme using the finite element method (FEM) in the spatial direction, offers an accurate approach for solving DO-FDEs. We examine stability and convergence analyses to validate the method’s applicability. Additionally, numerical experiments confirm theoretical results.

Keywords

Distributed-order fractional diffusion equation (DO-FDE) / Finite element method (FEM) / Grönwall inequality / Stability analysis / Convergence analysis / 65M12 / 65M60

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Funding

Iran National Science Foundation(4013841)

RIGHTS & PERMISSIONS

Shanghai University